Rotational magneto-photo-thermoacoustic dynamics in hydro-semiconductor media with mass diffusion

Abstract

Purpose: This study develops a comprehensive theoretical framework to investigate the propagation of photo-thermoacoustic waves in a magneto-hydro-semiconductor medium influenced by rotational motion and chemical diffusion. The work addresses the need for advanced modeling of multiphysical interactions in porous semiconductors subjected to external fields. Methods: The proposed model integrates the effects of rotation, magnetic fields, optical excitation, and mass diffusion within the framework of extended photo-thermoelasticity theory. Analytical solutions to the coupled governing equations are obtained using the normal mode technique. The analysis considers two-dimensional plane wave propagation in a porous semiconductor under harmonic excitation, with dimensionless variables used to generalize the results. Results: Key physical variables, temperature, stress, displacement, carrier density, chemical potential, and excess pore pressure, are evaluated and presented graphically under varying conditions of rotational speed and magnetic field intensity. The results reveal that both rotation and magnetic fields act as stabilizing agents, damping oscillations and reducing peak amplitudes across all fields. These effects significantly influence wave behavior, particularly in the presence of mass diffusion. Conclusion: The study demonstrates that the inclusion of rotational and magnetic field effects is essential for accurately modeling wave propagation in hydro-semiconductor materials. The proposed framework is particularly relevant for the design and optimization of devices such as optoelectronic sensors, thermoelectric converters, and photoacoustic imaging systems. Future extensions may incorporate nonlinear and nonlocal effects for enhanced predictive accuracy.

Keywords

photo-thermoacoustic rotational dynamics magnetic field hydro-semiconductors chemical diffusion coupled wave propagation

Introduction

The increasing integration of multiphysical effects in modern devices—such as optoelectronic sensors, semiconductor-based biosystems, and energy harvesting modules—has created a pressing need for more comprehensive models that capture the interactions between thermal, mechanical, optical, and electromagnetic fields. In particular, porous semiconductor materials exhibit complex behaviors when exposed to external influences like magnetic fields, rotational motion, and chemical diffusion. These phenomena significantly affect wave propagation, charge transport, and thermal stability in practical applications. Classical thermoelastic theories fall short in capturing these intricacies, especially in systems where mass diffusion, photo-excitation, and pore-pressure effects are dominant. Therefore, the current study aims to fill this critical modeling gap by proposing a rotational magneto-photo-thermoacoustic framework for hydro-semiconductors with chemical diffusion. This modeling approach is necessary to accurately predict and optimize the behavior of next-generation semiconductor devices operating in complex environments.

The dynamic behavior of semiconductor materials under the influence of coupled physical phenomena has garnered significant attention in recent years.¹ Among these, poro-semiconductors, characterized by their intricate porous structures, offer a unique medium for mass transport and wave propagation. These materials enable enhanced interaction between mechanical, thermal, and optical fields, providing a versatile platform for studying photo-thermoelastic effects.^2,3 Poro-semiconductors and hydrodynamics are intricately linked in the study of wave propagation in semiconductor materials, particularly when subjected to external stimuli such as magnetic and rotational fields.^4,5 In the framework of photo-thermoelasticity theory, the porous structure of semiconductors facilitates the diffusion of mass and chemical species, which becomes a critical factor in systems involving coupled thermal, mechanical, and optical interactions.⁶ The inclusion of hydrodynamic effects accounts for the fluid-like behavior of charge carriers and lattice vibrations, further enriched by the influence of magnetic fields that induce Lorentz forces and rotation fields that introduce Coriolis and centrifugal effects. Mass diffusion, governed by concentration gradients and chemical potential variations, serves as a driving force for enhanced transport phenomena, linked to thermal and mechanical responses.^7,8 Together, these interactions enable the design of advanced applications in biomedical devices, such as photoacoustic imaging and drug delivery systems, as well as in energy technologies, including thermoelectric materials and semiconductor-based sensors. The porous architecture not only facilitates enhanced mass transport and chemical diffusion but also enables intricate coupling between physical fields, making them an ideal candidate for studying the complex interplay of photo-thermoelastic effects.⁹ Understanding these interactions is crucial for designing next-generation devices that leverage semiconductor properties for high-performance applications.

Porous thermoelastic materials are specialized composite substances characterized by a network of interconnected voids or pores within a solid matrix.¹⁰ These materials exhibit unique thermo-mechanical properties due to their ability to absorb and transfer heat, coupled with their elastic response to mechanical loads.¹¹ Sherief and Hussein¹² developed a mathematical model for short-time filtration in poroelastic media, incorporating thermal relaxation and the two-temperature theory, offering refined predictions for heat and fluid transport in porous systems. Abousleiman and Ekbote¹³ presented analytical solutions for inclined boreholes in transversely isotropic porothermoelastic media, addressing stress and temperature fields under complex boundary conditions. Booker and Savvidou¹⁴ analyzed the consolidation process around a spherical heat source, highlighting the thermal-mechanical coupling in porous solids. Biot¹⁵ formulated a variational Lagrangian framework for non-isothermal, finite-strain mechanics of porous materials, accounting for thermomolecular diffusion effects, which laid the foundation for modern porothermoelastic theories. Hydrodynamic effects provide a deeper perspective on carrier dynamics within semiconductor materials. Traditional models, such as drift-diffusion equations, often fall short of capturing the nuanced behavior of charge carriers under high-field or nanoscale conditions.¹⁶ Hydrodynamic models, by contrast, treat electrons and holes as fluid-like entities, accounting for their collective behavior influenced by pressure, velocity gradients, and energy fluxes.¹⁷ This approach bridges the gap between microscopic quantum models and macroscopic continuum theories, offering a more comprehensive framework for analyzing charge and energy transport. In the context of poro-semiconductors, the integration of hydrodynamic principles allows for a robust examination of wave behavior under external influences.

Studying semiconductors within the framework of photo-thermoelasticity theory provides a comprehensive understanding of their coupled thermal, optical, and elastic responses under external stimuli.^18,19 This theory integrates the principles of photonic interactions, heat conduction, and elastic deformation, enabling the exploration of complex wave propagation phenomena in semiconductor media.^20,21 Over time, advancements have included the incorporation of nonlocal and fractional-order heat conduction models to address micro- and nanoscale effects.²² El-Sapa et al.²³ proposed a nonlocal excited semiconductor model using Moore–Gibson–Thompson theory and provided insights into its stability. Modern advancements in semiconductor wave modeling have incorporated additional multiphysical effects. Lotfy et al.²⁴ investigated variable thermal conductivity using fractional heat-magneto-photothermal theory for a semiconductor cavity. Lotfy et al.²⁵ extended the hyperbolic two-temperature theory to analyze photothermal excitation in magneto-thermoelastic semiconducting media. Such developments have been pivotal in enhancing the applicability of semiconductors in high-precision optoelectronic and photothermal systems, as well as in nanotechnology-driven applications, where the interplay of photo-thermal and elastic properties is crucial.^26,27 The inclusion of magnetic fields introduces additional complexities to wave propagation analysis. Magnetic fields, through the Lorentz force, impart anisotropic effects on charge carrier trajectories, significantly altering the mechanical and thermal wavefronts in semiconductor media.^28,29 This phenomenon forms the basis of several advanced technologies, including Hall-effect devices, magnetoresistive sensors, and spintronic systems.³⁰ Rotational fields, characterized by Coriolis and centrifugal forces, introduce dynamic perturbations that influence mechanical, thermal, and optical wave propagation, necessitating advanced mathematical modeling to capture their impact fully.^31–33

Classical and generalized thermoelasticity models have laid the foundation for current extensions. Biot³⁴ formulated thermoclasticity through irreversible thermodynamics. Lord and Shulman³⁵ introduced a generalized dynamical theory incorporating thermal relaxation. Green and Lindsay³⁶ developed a thermoelasticity theory addressing wave behavior under finite propagation speeds. Green and Naghdi^37,38 proposed models with undamped heat waves and revisited the fundamentals of thermomechanics, providing a key basis for modern photo-thermoelastic modeling. The extension of this theory to include hydrodynamic effects, magnetic fields, rotational dynamics, and chemical diffusion represents a significant leap forward in understanding the coupled behavior of elastic media.^39,40 Recent developments have extended traditional thermoelastic models by incorporating mass diffusion and semiconductor properties to account for more complex interactions in advanced materials.⁴¹ In particular, the interaction between thermal gradients, stress, and mass diffusion, such as in semiconductors, is crucial for understanding phenomena like thermal expansion, stress-induced diffusion, and the impact of temperature variations on carrier transport.^42,43 With the growing interest in nanostructured and microstructured materials, such as semiconductors, these models have been enhanced to incorporate the effects of nonlocality and fractional-order time derivatives to address scale-dependent behaviors and more accurately predict material responses at the nanoscale.⁴⁴ The interaction between thermoelasticity and mass diffusion, especially in semiconductor materials, is essential for understanding their performance in applications such as thermoelectrics, optoelectronics, and heat management in microelectronics.^45–48 These developments continue to push the boundaries of material design and optimization for cutting-edge technologies.

Chemical diffusion and mass transport are equally vital in understanding semiconductor behavior, particularly in porous structures where gradients in chemical potential drive the movement of species.^49–51 The coupling of chemical diffusion with thermoelastic effects creates a complex network of interactions that significantly influence the thermal, optical, and mechanical responses of the material.⁵² This aspect is particularly important in applications requiring precise control of transport phenomena, such as drug delivery systems, environmental sensing, and energy harvesting devices.⁵³ By incorporating chemical diffusion into the analysis, researchers can develop models that better reflect the real-world behavior of semiconductor systems.⁵⁴

In addition to analytical models, recent research has emphasized the role of high-fidelity computational techniques in capturing nonlinear thermoacoustic behaviors and unsteady heat-fluid interactions, particularly in energy conversion systems such as thermoacoustic engines. Notably, Guo et al.^55–57 have demonstrated the use of unsteady RANS, LES, and hybrid URANS/LES models to simulate turbulent flow structures, energy transport, and acoustic behavior with remarkable accuracy. While such methods are computationally intensive, they offer a valuable complement to analytical approaches, especially when nonlinear and turbulence effects dominate. The current study focuses on analytical insights into coupled multiphysical behavior in semiconductor media; however, integrating or benchmarking against CFD-based thermoacoustic models remains an important future direction.

The synergy of these phenomena opens new avenues for innovation in semiconductor design and functionality. By studying the combined effects of rotational and magnetic fields, chemical diffusion, and photo-thermoelasticity, this paper aims to provide a holistic understanding of wave propagation in magneto-hydro-semiconductor media. The analysis is conducted using advanced mathematical tools, including normal mode analysis, to derive analytical solutions that capture the intricate interplay of these effects. Graphical representations of key parameters, such as temperature distribution, displacement, carrier density, and diffusion concentration, offer valuable insights into the underlying physics and practical implications. The findings of this study are anticipated to have far-reaching implications in various fields, including the development of high-efficiency optoelectronic devices, precision sensors, and innovative energy systems. By addressing the complex interplay of multiple physical phenomena, this research contributes to the growing body of knowledge required to meet the challenges of modern materials science and engineering.

Mathematical model and basic equations

Model assumptions and limitations

In developing the present analytical model, several assumptions have been adopted to simplify the governing equations and enable tractable solutions. These include.

• The material is considered linear, isotropic, and homogeneous, with constant physical properties.

• The porous medium is assumed to be fully saturated and behaves according to linear poroelastic theory.

• A two-dimensional formulation is used, neglecting out-of-plane effects and edge interactions.

• A single relaxation time governs the thermal and electromagnetic fields.

• The model employs normal mode analysis, assuming harmonic time dependence and infinite spatial extent.

• Quantum and nonlocal effects, as well as nonlinearities in carrier transport and thermal response, are not considered.

While these assumptions provide a clear framework for analyzing the coupled thermo-magneto-acoustic behavior in hydro-semiconductor media, they may limit applicability in scenarios involving strongly nonlinear responses, anisotropic materials, nanoscale effects, or transient thermal loading. Future extensions could incorporate multi-scale modeling, non-Fourier heat conduction, and nonlocal or fractional-order formulations to better address such complexities.

Mathematical model

Considering the photo-thermoacoustic problem in a magneto-hydro-semiconductor medium with rotational dynamics. The following governing equations describe the coupled behavior of thermal, optical (charge carriers $N$ ), and mechanical fields during the diffusive material concentration C of the porous-semiconductor medium under the influence of external fields (magnetic and rotation fields).

(i) The hydrodynamic equation for charge carriers in a semiconductor, considering both the interaction between thermal and carrier density rate with the thermal activation coupling coefficient $κ$ , can be expressed as^30,40,41

\frac{\partial N}{\partial t} = D_{E} \nabla^{2} N - \frac{N}{τ} + κ T .

(1)

(ii)The equation of motion under the effect of both rotation and magnetic field describes the behavior of charge carriers in a magneto-hydrodynamic system. This equation incorporates the Lorentz force $\overset{⇀}{F}$ due to the magnetic field, the Coriolis force due to rotation ( $Ω$ is the angular velocity), and the standard inertial and mass diffusion. Below is the derivation and presentation of the equation of motion^36,42

(λ + μ) \nabla (\nabla \cdot \overset{⇀}{u}) + μ \nabla^{2} \overset{⇀}{u} - \nabla (α_{p} P + γ T + β_{1} C + δ_{n} N) + \overset{⇀}{F} = ρ (\ddot{\underline{u}} + \underline{Ω} \times (\underline{Ω} \times \underline{u}) + 2 \underline{Ω} \times \underline{\dot{u}}) .

(2)

(iii) The coupled thermal, concentration, carrier density, and mechanical equation can be derived using the theory of photo-thermoelasticity of in the semiconductor medium is presented as^31,36

K \nabla^{2} T - (\frac{\partial}{\partial t} + τ_{o} \frac{\partial^{2}}{\partial t^{2}}) (m T - γ T_{o} e + c T_{o} C) + \frac{E_{g}}{τ} N = 0 .

(3)

(iv) The porosity effects and pore fluid pressure in a porous-semiconductor, are governed by the governing equations of fluid flow and mass conservation. In the context of a magneto-hydro-semiconductor medium with rotation, the governing equations must include the effects of pore fluid pressure ( $P$ ) and the porosity $n_{0}$ (which reflects the void space available for the fluid)³⁶

b_{w} (α_{w} \frac{\partial T}{\partial t} - \frac{\partial e}{\partial t}) - ρ_{w} \frac{\partial^{2} e}{\partial t^{2}} + \nabla^{2} P = 0,

(4)

here

b_{w} = \frac{g ρ_{w}}{k_{d}}

and

m = n_{0} ρ_{w} C_{w} + (1 - n_{o}) ρ_{s} C_{s}

(v) The mass conservation equation for the porous hydro-semiconductor medium, accounting for both concentration diffusion and thermal gradients, is given by the generalized form⁴⁰

D_{c} β_{2} \nabla^{2} e + D_{c} c \nabla^{2} T + (\frac{\partial}{\partial t} + τ_{d} \frac{\partial^{2}}{\partial t^{2}}) C = D_{c} b_{c} \nabla^{2} C .

(5)

where

β_{2} = (3 λ + 2 μ) α_{c}

α_{c}

is the linear diffusion parameter,

b_{c}

is a parameter which it measure the diffusive effect,

τ_{d}

expresses the diffusion relaxation time, and

c

measure the impact of thermoelastic diffusion.

The mechanical (stress–strain) equation in the context of the photo-thermoelasticity theory during concentration diffusion is^32,36

{σ_{i}}_{I} = λ u_{r},_{r} {δ_{i}}_{I} + μ u_{I},_{i} + μ u_{i, I} - (α_{p} P + γ T + β_{2} C + δ_{n} N) {δ_{i}}_{I} .

(6)

Chemical diffusion, which describes the transport of species based on concentration gradients and chemical potential can be expressed as⁴⁰

E = - β_{2} e + b_{c} C - c (T - T_{0}) .

(7)

where

0 \leq α_{p} \leq 1

represents the poroelastic coupling coefficient (

α_{p} = 1

, fully saturated porous medium,

α_{p} = 0

, corresponding to a purely solid medium).

These Maxwell equations provide the framework for analyzing the electromagnetic behavior in complex systems like the magneto-hydro-semiconductor medium under the influence of rotation, thermal gradients, and mass diffusion. The provided set of equations incorporates the effects of an initial magnetic field $\overset{⇀}{H} = (0, H_{0}, 0)$ , an induced magnetic field $\overset{⇀}{h} = - H_{0} (0, e, 0)$ , current density $\overset{⇀}{J}$ , and electric field $\overset{⇀}{E}$ . The equations are expressed as follows

\begin{array}{l} \overset{⇀}{J} = c u r l \overset{⇀}{h} - ε_{0} \dot{\vec{E}}, c u r l \vec{E} = - μ_{0} \dot{\overset{⇀}{h}}, \\ \overset{⇀}{E} = - μ_{0} (\dot{\overset{⇀}{u}} ⥄ \times \overset{⇀}{H}), div \overset{⇀}{h} = 0, \\ \overset{⇀}{F} = μ_{0} (\overset{⇀}{J} \times \overset{⇀}{H}) \end{array}} .

(8)

In equation (7),

μ_{0}

represents the permeability and

ε_{0}

is the permittivity.

The governing equations for the problem in a two-dimensional (2D) ( $x, z$ )-plane under the influences of magnetic fields, rotation, and chemical diffusion can be derived by simplifying the general equations to 2D. The displacement vector in 2D can be expressed in the form $\vec{u} = (u, 0, w)$ , $u = u (x, z, t), w = w (x, z, t)$ ). In this case, governing the electric and magnetic fields in 2D are

\begin{array}{l} J_{x} = H_{0} \frac{\partial e}{\partial z} - μ_{0} H_{0} ε_{0} \ddot{w}, J_{z} = - H_{0} \frac{\partial e}{\partial x} + μ_{0} H_{0} ε_{0} \ddot{u}, J_{y} = 0 \\ F_{x} = μ_{0} H_{0}^{2} \frac{\partial e}{\partial x} - μ_{0}^{2} H_{0}^{2} ε_{0} \frac{\partial^{2} u}{\partial t^{2}}, F_{z} = μ_{0} H_{0}^{2} \frac{\partial e}{\partial z} - μ_{0}^{2} H_{0}^{2} ε_{0} \frac{\partial^{2} w}{\partial t^{2}} . \end{array}} .

(9)

In the context of a hydro-poroelastic semiconductor medium, coupled equations with stresses, rotation, and magnetic fields in 2D are⁴³

(λ + μ) \frac{\partial e}{\partial x} + μ (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial z^{2}}) - \frac{\partial}{\partial x} (α_{p} P + γ T + β_{1} C + δ_{n} N) + μ_{0} H_{0}^{2} \frac{\partial e}{\partial x} - μ_{0}^{2} H_{0}^{2} ε_{0} \frac{\partial^{2} u}{\partial t^{2}} = ρ (\frac{\partial^{2} u}{\partial t^{2}} - Ω^{2} u + 2 Ω \dot{w}),

(10)

(λ + μ) \frac{\partial e}{\partial z} + μ (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial z^{2}}) - \frac{\partial}{\partial x} (α_{p} P + γ T + β_{1} C + δ_{n} N) + μ_{0} H_{0}^{2} \frac{\partial e}{\partial z} - μ_{0}^{2} H_{0}^{2} ε_{0} \frac{\partial^{2} w}{\partial t^{2}} = ρ (\frac{\partial^{2} w}{\partial t^{2}} - Ω^{2} w - 2 Ω \dot{u}),

(11)

σ_{x x} = 2 μ \frac{\partial u}{\partial x} + λ e - (γ T + α_{p} P + β_{2} C + (3 λ + 2 μ) d_{n} N),

(12)

σ_{x z} = μ (\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}) .

(13)

Introducing dimensionless quantities in governing equations is a critical step to simplify the equations, reduce the number of parameters, and identify dominant effects in the system. Below is a general form for non-dimensionalizing quantities for this phenomenon

\begin{array}{l} \bar{N} = \frac{δ_{n} N}{2 μ + λ}, ({\bar{x}}_{i}, {\bar{u}}_{i}) = C_{0} ξ (x_{i}, u_{i}), (\bar{t}, \bar{τ}, {\bar{τ}}_{o}) = C_{0}^{2} ξ (t, τ, τ_{o}), Ω^{'} = \frac{Ω}{C_{0}^{2} ξ}, E^{'} = \frac{E}{β_{2}}, \\ \bar{T} = \frac{γ T}{2 μ + λ}, {\bar{σ}}_{i}_{j} = \frac{{σ_{i}}_{j}}{μ}, \bar{P} = \frac{P}{2 μ + λ}, ξ = \frac{m}{K}, C_{0}^{2} = \frac{λ + 2 μ}{ρ}, C^{'} = \frac{β_{2} C}{2 μ + λ} . \end{array}} .

(14)

Substitute the dimensionless variables (equation (14)) into the governing equations and divide through by the characteristic scales to make all terms dimensionless

(\nabla^{2} - ε_{1} - ε_{2} \frac{\partial}{\partial t}) N + ε_{3} T = 0,

(15)

\nabla^{2} u + (B^{2} + R_{H} - 1) \frac{\partial e}{\partial x} - B^{2} \frac{\partial}{\partial x} (P + T + N + C) = B^{2} (α \frac{\partial^{2} u}{\partial t^{2}} - Ω^{2} u + 2 Ω \frac{\partial w}{\partial t}),

(16)

\nabla^{2} w + (B^{2} + R_{H} - 1) \frac{\partial e}{\partial z} - B^{2} \frac{\partial}{\partial z} (P + T + N + C) = B^{2} (α \frac{\partial^{2} w}{\partial t^{2}} - Ω^{2} w - 2 Ω \frac{\partial u}{\partial t}),

(17)

\nabla^{2} T + ε_{7} N + (\frac{\partial}{\partial t} + τ_{o} \frac{\partial^{2}}{\partial t^{2}}) (T + ε_{8} e + ε_{o} C) = 0,

(18)

\nabla^{2} P = ε_{4} \frac{\partial e}{\partial t} + ε_{5} \frac{\partial T}{\partial t} + ε_{6} \frac{\partial^{2} e}{\partial t^{2}},

(19)

\nabla^{2} e + δ_{1} \nabla^{2} T + (\frac{\partial}{\partial t} + τ_{d} \frac{\partial^{2}}{\partial t^{2}}) δ_{2} C - δ_{3} \nabla^{2} C = 0,

(20)

σ_{x x} = 2 \frac{\partial u}{\partial x} + \frac{λ}{μ} e - B^{2} (T + P + C + N),

(21)

σ_{x z} = \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}

(22)

where

ε_{0} = \frac{c γ T_{o}}{K β_{2} ξ}, ε_{1} = \frac{1}{τ D_{E} ξ C_{0}^{2}}, ε_{2} = \frac{1}{D_{E} ξ C_{0}^{2}}, ε_{3} = \frac{κ δ_{n}}{D_{E} C_{0}^{4} γ ξ^{2}}, ε_{4} = \frac{b_{w}}{λ + 2 μ}, ε_{5} = - \frac{b_{w} α_{w}}{ρ ξ},

ε_{6} = - \frac{ρ_{w}}{ρ}, B^{2} = \frac{λ + 2 μ}{μ}, ε_{7} = \frac{γ E_{g}}{τ m δ_{n}}, ε_{8} = \frac{γ^{2} T_{0}}{m (λ + 2 μ)}, δ_{1} = \frac{c (2 μ + λ)}{γ β_{2}}, δ_{2} = \frac{(2 μ + λ)}{γ β_{2}}, δ_{2} = \frac{2 μ + λ}{C_{0} ξ D_{c} β_{2}^{2}}, δ_{3} = \frac{b_{c} (2 μ + λ)}{β_{2}^{2}}, α = 1 + \frac{μ_{0}^{2} H_{0}^{2} ε_{0}}{ρ}, R_{H} = \frac{μ_{0} H_{0}^{2}}{μ}, \nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial z^{2}} .

Helmholtz’s decomposition theorem can be decomposed into the sum of an irrotational (gradient) $Π (x, y, t)$ component and a solenoidal (divergence-free) $ψ (x, y, t)$ component. In this case, the displacement components can be decomposed into the form $u = \frac{\partial Π}{\partial x} + \frac{\partial ψ}{\partial z}$ and $w = \frac{\partial Π}{\partial z} - \frac{\partial ψ}{\partial x}$ . Substitute this decomposition into the wave equation (equation of motion in 2D), yields

(\nabla^{2} + ϑ_{2} Ω^{2} - ϑ_{1} \frac{\partial^{2}}{\partial t^{2}}) Π - ϑ_{2} (P + T + N + C) + 2 ϑ_{2} Ω \frac{\partial ψ}{\partial t} = 0,

(23)

(\nabla^{2} - B^{2} (\frac{\partial^{2}}{\partial t^{2}} - Ω^{2})) ψ - 2 Ω B^{2} \frac{\partial Π}{\partial t} = 0,

(24)

where

ϑ_{1} = \frac{α B^{2}}{B^{2} + R_{H}}, ϑ_{2} = \frac{B^{2}}{B^{2} + R_{H}}

Summary of governing equations

Table 1 provides a concise overview of the key governing equations and the corresponding physical assumptions employed in the proposed magneto-photo-thermoacoustic model.

Table 1.

Summary of governing equations and physical assumptions in the proposed model.

Equation type	Governing equation (Simplified form)	Physical phenomena captured	Key assumptions
Carrier transport	Equation (1): Hydrodynamic carrier diffusion equation	Carrier mobility, thermal activation	Semiconductor carriers modeled as fluid-like; linear coupling
Equation of motion	Equation (2): Includes Coriolis & Lorentz forces	Mechanical motion under rotation & magnetic field	Small deformations; isotropic medium; linear elasticity
Thermal conduction	Equation (3): Modified Fourier law with relaxation	Heat flow with memory effect	Single thermal relaxation time; linear temperature distribution
Fluid Mass balance	Equations (4) and (5): Mass conservation with pore pressure and chemical concentration	Mass diffusion in porous media	Fully saturated porous medium; constant porosity
Stress-strain relation	Equation (6): Linear elastic relation with coupling to temperature and diffusion fields	Stress generation from thermal and chemical effects	Small strains; linear poroelasticity
Chemical diffusion	Equation (7): Based on chemical potential gradient	Mass transfer driven by chemical potential gradients	Constant diffusion coefficient; isothermal chemical diffusion
Maxwell’s equations (simplified)	Equations (8) and (9): Electromagnetic field relations	Magnetic field interaction with semiconductor	Quasi-static EM fields; linear permittivity and permeability
Displacement potentials	Equations (23) and (24): Scalar and vector potentials from Helmholtz decomposition	Separation of longitudinal and transverse waves	2D assumption; infinite medium in x-direction
Assumed solution form	Equation (25): Normal mode (harmonic) solution	Analytical tractability of PDE system	Steady-state harmonic excitation; semi-infinite domain

Solution to the problem

The analytical solution in this study is derived using the normal mode technique, which assumes harmonic variations in time and space for all physical fields. While this method simplifies the mathematical treatment and offers clear insights into the coupled behavior of the system, it inherently assumes linearity, steady-state behavior, and infinite or semi-infinite domains. As such, it may not capture transient dynamics, localized disturbances, nonlinear interactions, or edge effects that may arise in practical or nanoscale systems. Additionally, more complex damping mechanisms and dispersive phenomena are beyond the scope of this approach. Despite these limitations, the normal mode technique remains a powerful tool for analyzing fundamental wave propagation trends in coupled multiphysics environments. Future work may employ numerical simulations or time-domain methods to address more realistic boundary conditions and dynamic loading scenarios. The normal mode method is a powerful analytical technique used to solve coupled dynamic field problems, such as photo-thermoelasticity, where interactions between hydro-elastic, thermal, electromagnetic, and diffusion fields occur. It transforms partial differential equations (PDEs) into ordinary differential equations (ODEs) by assuming solutions with specific harmonic forms for each field variable $F (x, z, t)$ ^31,36

F (x, z, t) = F^{*} (x) \exp (ω t + i b z) .

(25)

where

ω

i

F^{*} (x)

and

b

are the angular frequency, imaginary unit, amplitude function (dependent on), and the wave number in the

z

-axis direction, respectively. Substitute the normal mode solutions into the governing PDEs equations (15)–(20), (22), and (24), yields

(D^{2} - α_{1}) N^{*} + ε_{3} T^{*} = 0,

(26)

(D^{2} - α_{3}) T^{*} + ε_{7} N^{*} + α_{4} (D^{2} - b^{2}) Π^{*} + α_{7} C^{*} = 0,

(27)

(D^{2} - b^{2}) P^{*} - α_{5} (D^{2} - b^{2}) Π^{*} - α_{6} T^{*} = 0,

(28)

{(D^{2} - b^{2})}^{2} Π * + δ_{1} (D^{2} - b^{2}) T * - (δ_{3} D^{2} - α_{8}) C * = 0,

(29)

(D^{2} - α_{2}) Π^{*} - ϑ_{2} (T^{*} + P^{*} + N^{*} + C^{*}) + α_{9} ψ^{*} = 0,

(30)

(D^{2} - α_{10}) ψ^{*} - α_{11} Π^{*} = 0,

(31)

σ_{x x}^{*} = 2 D u^{*} + \frac{λ}{μ} e^{*} - B^{2} (T^{*} + P^{*} + C^{*} + N^{*}),

(32)

σ_{x z} = D w + i b u,

(33)

where

\frac{d}{d x} = D,

α_{4} = (ω + τ_{0} ω^{2}) ε_{8}

α_{5} = ω (ε_{4} + ω ε_{6})

α_{6} = ω ε_{5}

α_{7} = (ω + τ_{0} ω^{2}) ε_{0}

α_{8} = (1 + τ_{d} ω^{2}) ω δ_{2} + δ_{3} b^{2}

α_{1} = b^{2} + ε_{1} + ω ε_{2}, α_{2} = b^{2} + ω^{2}, α_{3} = b^{2} - ω - τ_{0} ω^{2}

α_{9} = 2 i ω ϑ_{2} Ω

α_{10} = b^{2} - B^{2} (ω^{2} - Ω^{2})

α_{11} = 2 i B^{2} Ω ω

Solving the above-coupled equations by the elimination technique between $N^{*} (x), T^{*} (x), Π^{*} (x), ψ^{*} (x)$ , $C^{*} (x)$ and $P^{*} (x)$ (using MATLAB 2024a), yields

(D^{12} - E_{1} D^{10} + E_{2} D^{8} - E_{3} D^{6} + E_{4} D^{4} - E_{5} D^{2} + E_{6}) F^{*} (x) = 0,

(34)

where the main coefficients of equation (30) can be obtained in the Appendix.

By factorizing, the coupled ODEs (30) can be put systematically as

\prod_{i = 1}^{6} (D^{2} - k_{i}^{2}) {N^{*}, T^{*}, e^{*}, C^{*}, P^{*}} (x) = 0 .

(35)

The quantities $k_{n}^{2} (n = 1, . . . ., 6)$ are distinct real-positive roots of the auxiliary equation (39). When $x \to \infty$ , the main solutions of the field variable can be reformulated in the following linear form

{Π^{*}, T^{*}, C^{*}, P^{*}, N^{*}, ψ^{*}} (x) = \sum_{n = 1}^{6} M_{n} {1, a^{*}, b^{*}, c^{*}, d^{*}, f^{*}} {e^{- k_{n} x}}_{,}

(36)

where the boundary conditions are used to determine the values of the unknown constants

M_{n}

(n = 1, . . . ., 6)

and

a^{*} = \frac{(k_{i}^{2} - α_{1}) (- b^{2} + k_{i}^{2}) (α_{5} α_{7} ϑ_{2} - α_{7} (k_{i}^{2} - α_{2} + ψ α_{9}) - (- b^{2} + k_{i}^{2}) α_{4} ϑ_{2})}{ϵ_{3} (- b^{2} + k_{i}^{2}) (α_{7} ϑ_{2} - ϵ_{7} ϑ_{2}) - (k_{i}^{2} - α_{1}) (α_{6} α_{7} ϑ_{2} + (- b^{2} + k_{i}^{2}) ((α_{3} - k_{i}^{2}) ϑ_{2} + α_{7} ϑ_{2}))},

b^{*} = \frac{- a^{*} k_{i}^{2} + a^{*} α_{3} + b^{2} α_{4} - k_{i}^{2} α_{4} - d^{*} ϵ_{7}}{α_{7}}, c^{*} = \frac{b^{2} α_{5} - k_{i}^{2} α_{5} - a^{*} α_{6}}{b^{2} - k_{i}^{2}}, d^{*} = - \frac{a^{*} ϵ_{3}}{k_{i}^{2} - α_{1}}, f^{*} = \frac{α_{11}}{k_{i}^{2} - α_{10}} .

To obtain the displacement components from the governing equations in coupled systems using equations (16) and (17) according to the normal mode technique, yields

u^{*} (x) = \sum_{n = 1}^{6} (- k_{n} + i b f^{*}) e^{- k_{n} x},

(37)

w^{*} = \sum_{n = 1}^{6} (i b + k_{n} f^{*}) M_{n} e^{- k_{n} x} .

(38)

The mechanical (stresses) components are determined by solving coupled field equations (32) and (33)

σ_{x x}^{*} = \sum_{n = 1}^{6} (2 k_{n} (k_{n} - i b f^{*}) + \frac{λ}{μ} (k_{n}^{2} - b^{2}) - B^{2} (a^{*} + c^{*} + d^{*})) M_{n} e^{- k_{n} x},

(39)

σ_{x z}^{*} = \sum_{i = 1}^{6} (2 i b k_{i} + (k_{i}^{2} + b^{2}) f^{*}) M_{i} e^{- k_{n} x} .

(40)

While the normal mode technique offers analytical tractability and valuable physical insight into steady-state, linear wave propagation problems, it inherently assumes harmonic time dependence and spatial periodicity. As a result, it is not suitable for capturing transient phenomena such as pulse responses, shock fronts, or wave reflections from finite boundaries. Additionally, nonlinearities arising from temperature-dependent properties, strain-rate effects, or strong carrier-field interactions are outside the method’s scope. These limitations restrict the model’s applicability to linear, time-harmonic, and spatially infinite or semi-infinite domains. Future extensions using time-domain numerical methods or nonlinear solvers are recommended to address these complexities and provide a more complete picture of dynamic behavior in realistic semiconductor environments.

Boundary conditions

For the phenomenon involving rotational photo-thermoacoustic dynamics in a magneto-hydro-semiconductor medium with chemical diffusion, the boundary conditions at the free surface $x = 0$ can be formulated for each of the physical fields involved (hydro-mechanical, thermal, plasma, and chemical). Below are some example boundary conditions applicable to obtain the unknown parameters $M_{n}$ .

(1) Thermal boundary conditions based on the temperature gradient describe how heat flows through the boundary of a material. When the boundary of a semiconductor or any material is exposed to a temperature gradient, heat conduction occurs according to Fourier’s law, where the heat flux $q$ is proportional to the negative of the temperature gradient. Mathematically, this condition is expressed as

{- K \frac{\partial T}{\partial x} |}_{x = 0} = q .

(41)

(2) Concentration boundary condition is fixed at the boundary, if no chemical diffusion occurs across the boundary, indicating that the concentration gradient is zero and there is no chemical transport across the boundary, impose

\frac{\partial C (0, z, t)}{\partial x} = 0 .

(42)

(3) A chemical potential boundary condition specifies the behavior of the chemical potential $E$ at the boundary of the system, which is essential for problems involving chemical diffusion and reaction processes. Depending on the physical scenario, for an impermeable boundary or in scenarios where no chemical transport occurs across the boundary

\frac{\partial E (0, z, t)}{\partial x} = 0,

(43)

indicating that the gradient of the chemical potential is zero (or constant at the boundary), and no flux occurs.

(4) Carrier density boundary conditions, in the context of a recombination process, are essential for modeling the behavior of charge carriers at the boundaries of a semiconductor medium. Recombination is the process where free electrons combine, effectively reducing the carrier density. At the boundary of the semiconductor, this process can be modeled by specifying a boundary condition that accounts for the rate of recombination or generation of carriers ( $N_{0}$ : the carrier concentration) at the surface

N (0, z, t) = N_{0} e^{- λ t} .

(44)

where

λ

is the decay constant (decaying heat parameter).

(5) Mechanical boundary conditions related to normal stress are crucial for determining how a material responds at its surface under external loads or constraints. In the context of a magneto-hydro-semiconductor medium, the normal stress boundary condition can be expressed as the force $Ω_{1}$ per unit area acting perpendicular to the boundary surface

σ_{x x} (0, z, t) = Ω_{1} (z, t) = - Ω_{1}^{*} \exp (ω t + i b z) .

(45)

(6) The free tangential stress boundary condition is used to describe a surface where no tangential forces act, allowing the material to deform freely along the boundary. For a boundary lying in the xz-plane, the tangential stress components $σ_{x z}$ (shear stresses) must vanish at the free surface. Mathematically, this condition can be expressed as

σ_{x z} (0, z, t) = 0 .

(46)

These conditions ensure that no tangential forces are exerted on the boundary surface, reflecting a physically free surface where shear stresses are absent.

By applying the conventional mode method to equations (45)–(48), a resulting system of equations was derived

\sum_{i = 1}^{6} a^{*} k_{i} M_{i} = \frac{q}{K},

(47)

\sum_{i = 1}^{6} \frac{- a^{*} k_{i}^{2} + a^{*} α_{3} + b^{2} α_{4} - k_{i}^{2} α_{4} - d^{*} ϵ_{7}}{α_{7}} k_{i} M_{i} = 0,

(48)

\sum_{i = 1}^{6} k_{i} (k_{i}^{2} + b^{2} + Γ_{1} b^{*} + Γ_{2} a^{*}) M_{i} = 0,

(49)

\sum_{i = 1}^{6} \frac{ε_{3}}{α_{1} - k_{i}^{2}} M_{i} = N_{0}^{*} \exp (- λ t),

(50)

\sum_{i = 1}^{6} (2 k_{i} (k_{i} - i b f^{*}) + \frac{λ}{μ} (k_{i}^{2} - b^{2}) - B^{2} (a^{*} + c^{*} + d^{*})) M_{i} = - Ω_{1}^{*},

(51)

\sum_{i = 1}^{6} (2 i b k_{i} + (k_{i}^{2} + b^{2}) f^{*}) M_{i} = 0,

(52)

where

Γ_{1} = \frac{b_{c} (2 μ + λ)}{β_{2}^{2}}

Γ_{2} = \frac{c (2 μ + λ)}{γ β_{2}}

In this case, the unknown parameters $M_{n} (n = 1, 2, 3, 4, 5, 6)$ are obtained by solving the resulting equations derived from the conventional mode method. These parameters typically include physical quantities such as displacement components, temperature, carrier density, and electromagnetic field components, which are influenced by factors like rotation, magnetic fields, and chemical diffusion.

Numerical results and discussions

This section presents the numerical results and discussion regarding the photo-thermoacoustic behavior of poro-silicon (PSi) material, which is a key component in understanding the coupled dynamics under various external influences. The numerical solutions are obtained by solving the governing equations for the system, incorporating the effects of rotation, magnetic fields, chemical diffusion, and thermal gradients. The influence of different parameters, such as the magnetic field strength and rotation, is examined. The results are analyzed through graphical representations that highlight the propagation of waves, changes in temperature, displacement fields, and carrier density variations. These results provide valuable insights into the interplay between mechanical, thermal, and electromagnetic fields in a PSi medium, offering a better understanding of the material’s behavior under real-world conditions, particularly for applications in sensors, energy harvesting, and optoelectronic devices. The physical constants in SI units for PSi medium are listed in Table 2.^43–46 Table 2 summarizes the physical constants used for the numerical evaluation of the coupled photo-thermoacoustic behavior in the porous silicon (PSi) medium. These include material properties such as density, thermal conductivity, heat capacity, deformation potentials, diffusion coefficients, and relaxation times. The density values correspond to both the solid semiconductor matrix and the pore fluid, enabling accurate modeling of poroelastic effects. Thermal parameters such as specific heat and conductivity determine the heat transport behavior, while mechanical constants like Lamé parameters govern elastic wave responses. The carrier diffusion coefficient and lifetime influence the behavior of charge transport, and the deformation potentials capture the coupling between mechanical strain and carrier concentration. These values are selected from existing literature on PSi materials to reflect realistic and experimentally validated conditions for semiconductor applications involving porous structures.

Table 2.

The physical constants of PSi medium.

Unit	Symbol	Value	Unit	Symbol	Value
$N / m^{2}$	$λ$ $μ$	$3.64 \times 1 0^{10}$ $5.46 \times 1 0^{10}$	$N$	$p$	$100$
$k g / m^{3}$	$ρ_{s}$	$2.3 \times 1 0^{3}$	$1 / c m^{3}$	$N_{0}$	$9.65 \times 1 0^{9}$
$K$	$T_{0}$	$800$	$k g / m^{3}$	$ρ_{w}$	$1 0^{3}$
$s$	$τ$	$5 \times 1 0^{- 5}$	$m / s$	$k_{d}$	$1 0^{- 8}$
$m^{3}$	$d_{n}$	$- 9 \times 1 0^{- 31}$	$J / k g K$	$C_{w}$	$4 \times 1 0^{3}$
$m^{2} / s$	$D_{E}$	$2.5 \times 1 0^{- 3}$	${{}^{o}C}^{- 1}$	$α_{w}$	$2 \times 1 0^{- 4}$
$e V$	$E_{g}$	$1.11$	$J / k g K$	$C_{s}$	$6 \times 1 0^{2}$
$K^{- 1}$	$α_{s}$	$4.14 \times 1 0^{- 6}$	$J / k g K$	$C_{e}$	$695$
$W m^{- 1} K^{- 1}$	$K$	$150$	$s$	$τ_{o}$	$0.0002$
$k g / m^{3}$	$ρ$	$2000$	$Ω_{1}^{*} = 0.5$	$θ_{1} = 300 K$	$n_{0} = 0.4$
$ω = ω_{0} + i ξ$	$ω_{0} = - 2$	$ξ = 0.001$	$i = \sqrt{- 1}$	$t = 0.02 s$	$b = 1$

Influence of the rotational field

Figure 1 displays the distribution of physical fields versus distance under two conditions: with and without the presence of a rotational field, while maintaining a constant magnetic field and thermal relaxation time. The rotational field introduces Coriolis and centrifugal forces, which modify the momentum of charge carriers and affect the mechanical deformation of the medium. Figure 1 showcases the wave propagation behavior of various physical fields (temperature ( $T$ ), concentration ( $C$ ), chemical potential ( $E$ ), excess pore water pressure ( $P$ ), normal stress ( $σ_{x x}$ ), and carrier density ( $N$ )) in a hydro-semiconductor medium for mass diffusion and $λ = 0.3$ . These fields are plotted as functions of distance ( $x$ ) under two conditions: with and without a rotational field, while accounting for the effects of a magnetic field with one relaxation time. The temperature field exhibits a noticeable peak near the origin in the non-rotating case, followed by a decaying oscillatory pattern. When rotation is introduced, the peak value diminishes, and the waveform becomes smoother. This attenuation is due to the inertial resistance imposed by the Coriolis force, which disrupts the direct propagation of thermal waves, reducing their amplitude and increasing spatial damping. The concentration and chemical potential fields display similar oscillatory decay. Without rotation, these fields show steep gradients and localized peaks, attributed to unrestricted carrier diffusion. Rotation introduces stabilizing inertia, which reduces the rate of diffusion by opposing the motion of mass carriers. Consequently, both concentration and chemical potential gradients are moderated, leading to a more uniform distribution. Excess pore pressure is also dampened in the rotating case. This is due to the coupling between fluid saturation in pores and the mechanical displacements affected by rotation. The reduced oscillations indicate enhanced mechanical stability under rotation. Likewise, the normal stress and carrier density fields exhibit smoother profiles with decreased magnitudes, confirming that rotational effects act as a mechanical stabilizer throughout the system. However, with rotation, the peaks and troughs are reduced, implying that rotational effects can mitigate mechanical stresses in the medium, potentially enhancing its structural stability. The carrier density profile follows a similar trend, with reduced oscillations and peak amplitude in the presence of the rotational field. This demonstrates that rotation moderates the electromagnetic field’s influence on carrier dynamics. The rotational field stabilizes all physical fields, reducing the intensity of oscillations and peak amplitudes. This suggests that rotation acts as a damping mechanism, improving the stability of wave propagation in the hydro-semiconductor medium. The magnetic field, combined with the single relaxation time, induces coupled dynamic behavior, where the interaction between thermal, mechanical, and carrier diffusion fields governs the wave propagation. These effects are crucial in practical applications such as semiconductor device stability, energy transport, and optoelectronic performance, where controlling oscillations and ensuring uniform field distribution are essential.

Figure 1.

Wave propagation of the main physical fields against distance in a hydro-semiconductor medium. The results are shown for two cases: without a rotational field (solid red line) and with a rotational field (dashed blue line), under the effect of a magnetic field and a single relaxation time.

Although the developed model includes a wide range of coupled physical effects, namely rotation, magnetic fields, and chemical diffusion, the current numerical results are focused primarily on analyzing the influence of rotational and magnetic fields. The role of chemical diffusion, while present in the theoretical formulation, was not explicitly isolated in the graphical analysis. This simplification was made to ensure the interpretability of the core results; however, it represents a limitation in the data analysis. Future work will address this by conducting a comprehensive parametric study on chemical diffusion parameters, including the diffusion relaxation time and concentration gradients, and their interaction with thermal, mechanical, and electromagnetic responses in the hydro-semiconductor medium.

It is noted that although the developed model accounts for chemical diffusion through coupled concentration and chemical potential fields, the current numerical results do not explicitly isolate its effect. This is due to the focus on showcasing the impact of rotational and magnetic fields, which are more dominant in the present physical context. As a result, trends associated with chemical diffusion, such as delayed diffusion fronts or altered pressure fields, are embedded but not independently visualized. Future work will include a dedicated parametric analysis to investigate how varying diffusion coefficients, relaxation times, and chemical gradients independently influence the propagation and attenuation of thermoacoustic and mechanical waves.

Influence of the magnetic field

Figure 2 illustrates the wave propagation behavior of several physical fields (temperature ( $T$ ), concentration ( $C$ ), chemical potential ( $E$ ), excess pore water pressure ( $P$ ), normal stress ( $σ_{x x}$ ), and carrier density ( $N$ )) in a hydro-semiconductor medium. Figure 2 illustrates the physical fields under two scenarios: with and without the magnetic field, keeping rotation and thermal relaxation constant. The magnetic field introduces Lorentz forces, which exert a directional influence on moving charge carriers. This force modifies the carrier velocity and, in turn, influences the associated electric and thermal fields. In the absence of a magnetic field, sharp peaks and pronounced oscillations are observed across all physical variables. Upon introducing the magnetic field, these oscillations are significantly damped. For example, the temperature field experiences a reduction in peak amplitude due to the energy conversion between thermal and electromagnetic domains. The magnetic field induces electromagnetic coupling that absorbs part of the energy from the thermal wave, smoothing its profile. The carrier concentration becomes more stable under the influence of the magnetic field. The reduction in carrier mobility, caused by the Lorentz force restricting free motion, leads to slower diffusion and a more homogeneous density distribution. Similarly, the chemical potential becomes less steep, indicating a suppression of electrochemical gradients. The reduction in excess pore water pressure and mechanical stress further supports the magneto-mechanical coupling effect, where electromagnetic forces resist abrupt mechanical changes. Overall, the magnetic field enhances wave stability and reduces spatial fluctuations across all fields, which is desirable in device design for minimizing signal distortion and energy loss. The normal stress wave shows significant oscillations in both cases. With the magnetic field, the oscillations are less intense, and the peak stress values are reduced, indicating the magnetic field’s role in moderating mechanical disturbances. The carrier density follows a similar trend, with reduced peaks and smoother oscillations under the influence of the magnetic field. This shows that the magnetic field counteracts electromagnetic disturbances, resulting in more uniform carrier distributions. The presence of a magnetic field significantly alters the propagation dynamics of all the studied physical fields. The magnetic field introduces a stabilizing effect, reducing the amplitude and oscillatory nature of the waves. This behavior can be attributed to the coupling between electromagnetic and mechanical fields, which regulates the energy transfer and reduces gradients driving the wave dynamics. The damping effects of the magnetic field are crucial for applications requiring controlled wave propagation, such as in semiconductor devices, energy transport systems, and optoelectronics. These results emphasize the importance of magnetic fields in enhancing the stability and efficiency of hydro-semiconductor materials.

Figure 2.

Wave propagation of the main physical fields against distance in a hydro-semiconductor medium. The results are shown for two cases: without a magnetic field (solid red line) and with a magnetic field (dashed blue line), under the effect of a rotational field and a single relaxation time.

While the current study presents a coupled multiphysical model involving thermal, optical, mechanical, and diffusion fields under magnetic and rotational influences, the numerical results focus on a few illustrative cases and do not yet establish general or universal laws describing field interactions. The observed physical behavior is therefore local, highlighting specific effects of rotation or magnetic fields rather than providing comprehensive parametric trends. This represents an important limitation of the current work. In future research, a broader parametric sensitivity study will be performed to extract generalized field interaction trends. Additionally, the development of analytical scaling laws or data-driven models may help formulate more universal principles governing coupled wave propagation in porous semiconductor media.

While the present work focuses primarily on analytical modeling and qualitative trend evaluation, a sensitivity analysis was performed by comparing the system’s behavior under different values of magnetic and rotational field intensities. These comparisons, illustrated in Figures 1 and 2, reveal consistent damping effects and reduced peak responses, confirming the model’s ability to capture the physical influence of external fields. Although a full quantitative sensitivity study was beyond the current scope, the results indicate stable and physically consistent behavior across varying parameter conditions. Future research will aim to implement more comprehensive parametric and uncertainty analyses to further validate the model under broader operating scenarios.

Model comparison and performance analysis

To illustrate the scope and advantages of the proposed model, a comparison with traditional thermoelastic theories is presented in Table 3. The proposed model captures a broader range of coupled physical interactions, including magnetic fields, rotational effects, and mass diffusion, not addressed by classical frameworks. While traditional models are suitable for simplified, non-porous systems, the present model offers enhanced predictive capability for modern applications involving porous semiconductors and multiphysics environments.

Table 3.

Comparison of the proposed model with traditional thermoelastic models.

Feature/Scenario	Proposed model (present Study)	Classical thermoelastic Models (e.g., Lord–Shulman, Green–Naghdi)	Remarks
Magnetic field effect	✔ Included (via Lorentz force terms)	✘ Typically neglected or weakly coupled	Crucial for semiconductor sensors, spintronics, and magneto-mechanical systems
Rotational dynamics	✔ Included (Coriolis and centrifugal effects)	✘ Not considered	Important for rotating systems or gyroscopic sensor modeling
Chemical/Mass diffusion	✔ Modeled via concentration and chemical potential equations	✘ Usually absent	Relevant for energy storage, bio-sensing, and porous media
Carrier density (Optical field)	✔ Explicitly included and coupled	✘ Typically not modeled	Enhances applicability to photoacoustic semiconductor media
Porosity/Hydro effects	✔ poroelastic formulation and excess pore pressure included	✘ Most classical models assume solid, non-porous media	Suitable for porous silicon (PSi) and nanostructured semiconductors
Solution method	Analytical via normal mode analysis	Analytical (often Laplace/Fourier transform-based)	Both are closed-form methods, but differ in boundary treatment and assumptions
Computational complexity	Moderate to high (due to coupled PDEs and multiple physical fields)	Low to moderate	Tradeoff between physical accuracy and simplicity
Transient/Localized heat effects	✘ Limited (harmonic time dependence assumed)	✔ Some include pulse-like or step function sources	Proposed model may underperform in highly transient scenarios
Nonlinearity and Nanoscale effects	✘ Not included (linear, continuum-based)	✘ Also not included unless specially modified	Both types of models require further extension to address nonlinear or quantum-scale effects
Best use cases	Coupled photo-magneto-thermoacoustic analysis in porous, rotating semiconductors with mass diffusion	Simple thermal–mechanical wave propagation in homogeneous, non-rotating solids	Proposed model excels in multiphysics and semiconductor-specific environments

Although a direct quantitative comparison with existing analytical or numerical models is outside the current scope, the qualitative trends observed in our simulations align with previously reported results in thermoelastic and semiconductor literature. For instance, the damping effects of magnetic fields on thermal and carrier wave amplitudes are consistent with findings reported in works employing Green–Naghdi or Lord–Shulman frameworks.^25,27,45 Similarly, the inertial stabilization introduced by rotational motion mirrors outcomes described in models involving Coriolis effects in rotating semiconductors.^7,30,48 The reduction in oscillatory behavior and smoothing of stress and concentration profiles are well-aligned with previously observed stabilization effects in magneto-thermoelastic porous media. These comparisons qualitatively validate the reliability and physical consistency of the proposed multiphysics model.

The results of this study extend classical models by simultaneously accounting for rotational motion, magnetic fields, and chemical diffusion in a porous semiconductor medium. While a direct numerical comparison with earlier models was not conducted, the observed damping effects, field stabilization, and redistribution trends qualitatively agree with previous results and offer enhanced predictive capabilities under multiphysics coupling. The proposed model is particularly suited for environments involving strong rotational and electromagnetic influences, where traditional models may not fully capture the interactive effects on wave behavior. Future work will focus on benchmarking the present model against simplified and fractional-order counterparts to quantify the performance improvements under various physical regimes.^58,59

A clear distinction in wave behavior is observed between field-free and field-influenced environments. In the absence of rotational or magnetic fields, the physical parameters, such as temperature, stress, and carrier density, exhibit higher peak amplitudes and more pronounced oscillatory behavior. The introduction of a rotational field dampens these oscillations and reduces the amplitude, acting as an inertial stabilizer. Similarly, the magnetic field modifies wave characteristics through Lorentz coupling, leading to smoother, lower-magnitude profiles and altered propagation speed. These differences emphasize the critical role of external fields in controlling wave energy distribution and stability in semiconductor materials (Table 4).

Table 4.

Differences in wave behavior with and without external fields.

Parameter	Without rotation/Magnetic field	With rotation/Magnetic field	Effect explanation
Temperature	Higher peaks, faster decay, sharp oscillations	Lower peaks, gradual decay, smoother profile	Lorentz force suppresses thermal wavefronts; rotation adds inertia
Concentration	Steeper gradients and localized spikes	Smoothed diffusion curve, reduced oscillations	Field coupling resists sharp diffusion transitions
Chemical potential	Pronounced fluctuations and local maxima	Damped and stable profile	Stabilized due to reduced energy gradients
Excess pore pressure	High oscillation amplitudes	Attenuated pressure variations	Rotational damping and magnetic resistance affect pore fluid dynamics
Stress	Large oscillations, frequent peak-trough transitions	Lower amplitude, smoother stress wave	Rotation reduces mechanical wave intensity
Carrier density	Higher mobility, sharp peaks, and diffusion fronts	Lower amplitude, stabilized carrier distribution	Magnetic field reduces mobility and controls carrier transport

Conclusions

This study has presented an analytical framework for modeling photo-thermoacoustic wave propagation in a porous silicon (PSi) medium under the combined influences of magnetic fields, rotational dynamics, and chemical diffusion. The results derived using the normal mode technique reveal that both rotational and magnetic fields significantly dampen oscillatory behaviors in key physical fields such as temperature, displacement, carrier density, and stress. Specifically, the presence of a magnetic field reduces peak amplitudes in temperature and carrier waves, while the rotational field stabilizes mechanical and chemical responses by minimizing excess pressure and stress oscillations.

The coupled nature of thermal, electromagnetic, and diffusion effects observed in the study highlights the strong interplay between external field modulation and wave behavior in semiconductor media. These insights are critical for the design of next-generation porous semiconductors used in high-performance optoelectronic devices, magneto-sensitive sensors, and thermal management systems. For example, the ability to suppress carrier mobility and heat fluctuations through magnetic or rotational field control could improve the stability and signal quality in photoacoustic imaging systems or MEMS-based biosensors.

Furthermore, the inclusion of chemical diffusion modeling provides a pathway for applying the findings to biomedical platforms such as drug-delivery systems, where controlled mass transport is required alongside temperature and stress regulation. Future work may extend this model to include nonlocal effects, nonlinearities, or layered material geometries to more closely reflect nanostructured semiconductor environments.

Overall, the study contributes to a deeper understanding of multiphysics interactions in porous semiconductor media and offers predictive tools for tuning wave behavior through external fields to meet the needs of real-world applications.

The theoretical insights gained from this study have direct relevance to several emerging technologies. In micro-electro-mechanical systems (MEMS), the ability to control thermoacoustic and mechanical wave behavior through rotational and magnetic fields can significantly enhance signal stability and thermal management. In biomedical sensing, porous semiconductor platforms that respond predictably to photo-thermal and chemical stimuli are crucial for drug delivery and bio-detection applications. Additionally, the stabilization effects observed under magnetic and rotational influence offer advantages in the design of optoelectronic systems, such as infrared detectors and photoacoustic imaging devices, where minimizing wave distortion is essential. These findings position the proposed model as a foundational tool for engineering next-generation semiconductor devices with improved multiphysical performance.

Applications and future work

The results of this study offer direct applicability to the engineering of semiconductor devices subjected to complex multiphysics environments. In particular, photoacoustic sensors, magneto-thermoelectric systems, and porous semiconductor biosensors can benefit from the understanding of how rotational and magnetic fields modulate thermal, mechanical, and carrier-related behavior. The model provides a predictive framework for optimizing device stability, minimizing wave-induced noise, and enhancing energy conversion efficiency. Additionally, the chemical diffusion dynamics captured in this model are relevant for biomedical and microfluidic devices where mass transport is coupled with heat and stress responses. These insights facilitate the rational design of advanced materials and devices operating in demanding conditions such as aerospace platforms, high-field electronics, and smart sensor technologies.

While the proposed model provides deep insights into the coupled field dynamics in hydro-semiconductor media, several challenges must be addressed for practical implementation in semiconductor manufacturing. Real-world materials often exhibit anisotropy, nano-scale nonlocal behavior, and layered geometries not fully captured by the current formulation. Additionally, some model parameters, such as chemical diffusion coefficients or rotation-induced stresses, are difficult to measure or control with high precision during fabrication. Integrating the model with advanced simulation platforms and validating it through experimental calibration will be essential steps toward translating these theoretical results into manufacturing and device optimization processes. Nonetheless, the present model offers a robust foundation for understanding multiphysical effects that are increasingly relevant in next-generation semiconductor design.

The multifield coupling framework developed in this study offers strong potential for interdisciplinary collaboration across various technological domains. In materials science, partnerships could support the fabrication and testing of porous, thermomagnetic semiconductors that align with model predictions. In photonics and optoelectronics, the model can aid in optimizing thermal and mechanical stability under high-frequency excitation. Mechanical and aerospace engineers may leverage the findings to improve rotating or field-exposed components in harsh environments. Biomedical applications also stand to benefit, particularly in porous materials used for sensing or targeted drug transport. Finally, collaboration with computational scientists could enable hybrid analytical-numerical extensions of this model, integrating turbulent flow or quantum transport phenomena. These cross-disciplinary connections highlight the versatility and broader impact of the presented research.

While the model developed in this work provides a detailed multiphysical formulation and numerical illustrations, the structural analysis of the solutions remains limited. Specifically, the regularity, asymptotic behavior, and mathematical stability of the coupled fields (e.g., temperature, carrier density, and stress) were not fully characterized. Furthermore, the absence of a generalized influence law for field interactions restricts the current conclusions to case-specific insights. Future research will focus on deriving more rigorous analytical results, such as scaling relationships, parametric bifurcations, or asymptotic limits, to better understand the global behavior of the system. Such analysis will enhance the theoretical foundation of the model and promote broader applicability to physical and engineering systems.

Footnotes

Acknowledgments

The project was funded by KAU Endowment (WAQF) at King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge with thanks WAQF and the Deanship of Scientific Research (DSR) for technical and financial support.

ORCID iD

Khaled Lotfy

Author contributions

All authors have equally participated in the preparation of the manuscript during the implementation of ideas, findings results, and writing of the manuscript.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The Current submission does not contain the pool data of the manuscript, but the data used in the manuscript will be provided on request.*

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Appendix

(53)

Θ_{1} = - δ_{3}^{- 1} {α_{8} - δ_{1} (α_{7} + α_{4} ϑ_{2}) + δ_{3} (b^{2} + α_{1} + α_{2} + α_{3} + α_{10}) + (α_{5} - α_{4} + α_{5}) δ_{3} ϑ_{2}},

Θ_{2} = δ_{3}^{- 1} (- α_{8} (b^{2} + α_{1} + α_{2} + α_{3} + α_{10}) + α_{7} δ_{1} (2 b^{2} + α_{1} + α_{2} + α_{10})

- (α_{1} + α_{2} + α_{3} + α_{10}) b^{2} δ_{3} - (α_{2} + α_{3} + α_{10}) α_{1} δ_{3} - α_{2} δ_{3} (α_{3} + α_{10})

- (α_{3} α_{10} + α_{9} α_{11} - ϵ_{3} ϵ_{7}) δ_{3} - (2 b^{2} + α_{1} + α_{3}) α_{5} ϑ_{2} + α_{4} (α_{8} - α_{6}) ϑ_{2}

- α_{5} (α_{7} + α_{8} + α_{10}) ϑ_{2} + (3 b^{2} + α_{1} + α_{10}) α_{4} δ_{1} ϑ_{2} + α_{5} α_{7} δ_{1} ϑ_{2} - α_{6} α_{7}

(54)

+ (2 b^{2} + α_{1} - α_{6}) α_{4} δ_{3} ϑ_{2} - (b^{2} + α_{1} + α_{3}) α_{5} δ_{3} ϑ_{2} + δ_{3} ϑ_{2} (α_{4} α_{10} - α_{5} α_{10} + α_{4} ϵ_{3})),

Θ_{3} = - δ_{3}^{- 1} ((b^{2} + α_{1} + α_{2}) α_{6} α_{7} + (b^{2} + α_{2}) α_{1} α_{8} + b^{2} (α_{2} + α_{3}) α_{8} + (α_{1} + α_{2}) α_{3} α_{8} + α_{6} α_{7} α_{10}

+ (b^{2} + α_{1} + α_{2} + α_{3}) α_{8} α_{10} + α_{8} α_{9} α_{11} - α_{7} δ_{1} (b^{4} + 2 b^{2} (α_{1} + α_{2}) + α_{1} α_{2} + α_{9} α_{11} + (2 b^{2} + α_{1} + α_{2}) α_{10})

+ α_{1} b^{2} (α_{2} + α_{3} + α_{10}) δ_{3} + (b^{2} + α_{1}) α_{2} α_{3} δ_{3} + (b^{2} + α_{1}) α_{2} α_{10} δ_{3} + (b^{2} + α_{1} + α_{2}) α_{3} α_{10} δ_{3}

+ (b^{2} + α_{1} + α_{3}) α_{9} α_{11} δ_{3} - ϵ_{3} ϵ_{7} (α_{8} + (b^{2} + α_{2} + α_{10}) δ_{3}) + b^{2} (b^{2} + 2 α_{1} + 2 α_{3} - 2 α_{7} δ_{1}) α_{5} ϑ_{2} + α_{1} α_{3} α_{5} ϑ_{2}

+ (2 b^{2} + α_{1} + α_{8} + α_{10}) α_{4} α_{6} ϑ_{2} + α_{5} α_{7} ϑ_{2} (2 b^{2} + α_{1} + α_{10}) - (2 b^{2} + α_{1} + α_{10}) α_{4} α_{8} ϑ_{2} - (3 b^{2} + α_{1}) α_{4} α_{10} δ_{1} ϑ_{2}

+ (2 b^{2} + α_{1} + α_{3}) α_{5} α_{10} ϑ_{2} - 3 b^{2} (b^{2} + α_{1}) α_{4} δ_{1} ϑ_{2} - (2 b^{2} + α_{1}) α_{5} α_{7} δ_{1} ϑ_{2} + (b^{2} + α_{1} + α_{3} + α_{10}) α_{5} α_{8} ϑ_{2}

(55)

- ((α_{7} - b^{2}) α_{5} α_{10} + b^{4} α_{4} + 2 b^{2} (α_{1} + α_{10}) α_{4} - b^{2} (α_{1} + α_{3}) α_{5} + α_{1} (α_{4} α_{10} - α_{3} α_{5}) - (b^{2} + α_{1}) α_{4} α_{6}) δ_{3} ϑ_{2} + ((1 + α_{3}) α_{5} + α_{4} α_{6}) α_{10} δ_{3} ϑ_{2} + ϵ_{3} ϑ_{2} (α_{5} α_{7} - α_{4} α_{8} - α_{4} δ_{3} (2 b^{2} + α_{10}) - α_{5} ϵ_{7} (1 + δ_{3}))),

Θ_{4} = δ_{3}^{- 1} (- b^{2} (α_{1} + α_{2} + α_{10}) α_{6} α_{7} - α_{1} α_{2} (α_{6} α_{7} + b^{2} α_{8}) - (b^{2} (α_{1} + α_{2} + α_{10}) + α_{1} α_{2}) α_{3} α_{8}

- (α_{1} + α_{2}) α_{6} α_{7} α_{10} - (b^{2} (α_{1} + α_{2}) + α_{1} α_{2}) + (α_{1} + α_{2}) α_{3}) α_{8} α_{10} - α_{1} (α_{3} α_{5} + α_{4} α_{6}) α_{10} ϑ_{2}

- (α_{6} α_{7} + (b^{2} + α_{1} + α_{3}) α_{8}) α_{9} α_{11} + b^{4} (α_{1} + α_{2} + α_{10}) α_{7} δ_{1} + 2 b^{2} α_{7} δ_{1} (α_{1} α_{2} + (α_{1} + α_{2}) α_{10})

+ α_{7} (α_{1} α_{2} α_{10} + (2 b^{2} + α_{1}) α_{9} α_{11}) δ_{1} - b^{2} (α_{1} α_{2} (α_{3} + α_{10}) + (α_{1} + α_{2}) α_{3} α_{10} + (α_{1} + α_{3}) α_{9} α_{11}) δ_{3}

- α_{1} α_{3} (α_{2} α_{10} + α_{9} α_{11}) δ_{3} + ϵ_{3} ϵ_{7} ((b^{2} + α_{2} + α_{10}) α_{8} + b^{2} (α_{2} + α_{10} + α_{2} α_{10} + α_{9} α_{11}) δ_{3}

- b^{4} (α_{1} + α_{3}) α_{5} ϑ_{2} - b^{4} (α_{4} (α_{6} - α_{8}) + α_{5} (α_{7} + α_{10})) ϑ_{2} - 2 b^{2} (α_{3} α_{5} + α_{4} α_{6} + α_{1} (α_{5} α_{7} - α_{4} α_{8})) ϑ_{2}

- b^{2} (α_{1} α_{8} + α_{3} α_{8} + α_{8} α_{10}) α_{5} ϑ_{2} - (α_{1} α_{3} α_{5} + (b^{2} + α_{1}) α_{4} α_{6}) α_{8} ϑ_{2} - 2 b^{2} (α_{1} + α_{3}) α_{5} α_{10} ϑ_{2}

- 2 b^{2} (α_{4} α_{6} + α_{5} α_{7} + α_{4} α_{8}) α_{10} ϑ_{2} - α_{1} (α_{5} α_{7} + α_{4} α_{8}) α_{10} ϑ_{2} - α_{5} α_{8} (α_{1} ϑ_{2} + α_{3} α_{5}) α_{8} α_{10} ϑ_{2}

- α_{4} (α_{6} α_{8} α_{10} + b^{6} δ_{1}) ϑ_{2} + b^{4} (3 α_{1} α_{4} + α_{5} α_{7} + 3 α_{4} α_{10}) δ_{1} ϑ_{2} + b^{2} (2 α_{1} α_{5} α_{7} + 3 α_{1} α_{4} α_{10} + 2 α_{5} α_{7} α_{10}) δ_{1} ϑ_{2}

+ α_{1} α_{5} α_{7} α_{10} δ_{1} ϑ_{2} + b^{2} (α_{4} (b^{2} α_{1} - α_{1} α_{6} + b^{2} α_{10}) - α_{1} α_{3} α_{5} + (2 α_{1} α_{4} - α_{1} α_{5} - α_{3} α_{5}) α_{10}) δ_{3} ϑ_{2}

(56)

- α_{10} (α_{1} α_{3} α_{5} + (b^{2} + α_{1}) α_{4} α_{6}) δ_{3} ϑ_{2} - ϵ_{3} ϑ_{2} (2 b^{2} (α_{5} α_{7} - α_{4} α_{8}) + (α_{5} α_{7} + α_{4} α_{8}) α_{10} - b^{2} α_{4} δ_{3} (b^{2} + 2 α_{10}) - 2 b^{2} α_{5} ϵ_{7}) + ϵ_{3} ϵ_{7} ϑ_{2} (α_{5} α_{8} + α_{5} α_{10} + α_{5} δ_{3} (b^{2} + α_{10}))),

Θ_{6} = δ_{3}^{- 1} (- b^{2} α_{1} α_{2} (α_{6} α_{7} + α_{3} α_{8}) α_{10} - b^{2} α_{1} (α_{6} α_{7} + α_{3} α_{8}) α_{9} α_{11} + b^{4} α_{1} α_{7} (α_{2} α_{10}

+ α_{9} α_{11}) δ_{1} - b^{2} ϵ_{3} ϵ_{7} (α_{2} α_{10} + α_{9} α_{11}) α_{8} - b^{4} α_{1} (α_{3} α_{5} + α_{4} α_{6} + α_{5} α_{7} + α_{4} α_{8}) α_{10} ϑ_{2}

(57)

- b^{2} α_{1} (α_{3} α_{5} + α_{4} α_{6}) α_{8} α_{10} ϑ_{2} + b^{4} α_{1} (b^{2} α_{4} + α_{5} α_{7}) α_{10} δ_{1} ϑ_{2} - b^{4} ϵ_{3} ϑ_{2} α_{10} (α_{5} α_{7} - α_{4} α_{8} + α_{5} ϵ_{7} + α_{5} α_{8} ϵ_{7})) .

Θ_{5} = - δ_{3}^{- 1} (b^{2} α_{1} α_{2} α_{6} α_{7} + b^{2} α_{1} α_{2} α_{3} α_{8} + b^{2} α_{1} α_{6} α_{7} α_{10} + b^{2} α_{2} α_{6} α_{7} α_{10} + α_{1} α_{2} α_{6} α_{7} α_{10} + b^{2} α_{1} α_{2} α_{8} α_{10}

+ b^{2} α_{1} α_{3} α_{8} α_{10} + b^{2} α_{2} α_{3} α_{8} α_{10} + α_{1} α_{2} α_{3} α_{8} α_{10} + b^{2} α_{6} α_{7} α_{9} α_{11} + α_{1} α_{6} α_{7} α_{9} α_{11} + b^{2} α_{1} α_{8} α_{9} α_{11}

+ b^{2} α_{3} α_{8} α_{9} α_{11} + α_{1} α_{3} α_{8} α_{9} α_{11} - b^{4} α_{1} α_{2} α_{7} δ_{1} - b^{4} α_{1} α_{7} α_{10} δ_{1} - b^{4} α_{2} α_{7} α_{10} δ_{1} - 2 b^{2} α_{1} α_{2} α_{7} α_{10} δ_{1}

- b^{4} α_{7} α_{9} α_{11} δ_{1} - 2 b^{2} α_{1} α_{7} α_{9} α_{11} δ_{1} + b^{2} α_{1} α_{2} α_{3} α_{10} δ_{3} + b^{2} α_{1} α_{3} α_{9} α_{11} δ_{3} - b^{2} α_{2} α_{8} ϵ_{3} ϵ_{7}

- b^{2} α_{8} α_{10} ϵ_{3} ϵ_{7} - α_{2} α_{8} α_{10} ϵ_{3} ϵ_{7} - α_{8} α_{9} α_{11} ϵ_{3} ϵ_{7} - b^{2} α_{2} α_{10} δ_{3} ϵ_{3} ϵ_{7} - b^{2} α_{9} α_{11} δ_{3} ϵ_{3} ϵ_{7} + b^{4} α_{1} α_{3} α_{5} ϑ_{2}

+ b^{4} α_{1} α_{4} α_{6} ϑ_{2} + b^{4} α_{1} α_{5} α_{7} ϑ_{2} - b^{4} α_{1} α_{4} α_{8} ϑ_{2} + b^{2} α_{1} α_{3} α_{5} α_{8} ϑ_{2} + b^{2} α_{1} α_{4} α_{6} α_{8} ϑ_{2} + b^{4} α_{1} α_{5} α_{10} ϑ_{2}

+ b^{4} α_{3} α_{5} α_{10} ϑ_{2} + 2 b^{2} α_{1} α_{3} α_{5} α_{10} ϑ_{2} + b^{4} α_{4} α_{6} α_{10} ϑ_{2} + 2 b^{2} α_{1} α_{4} α_{6} α_{10} ϑ_{2} + b^{4} α_{5} α_{7} α_{10} ϑ_{2} + 2 b^{2} α_{1} α_{5} α_{7} α_{10} ϑ_{2}

- b^{4} α_{4} α_{8} α_{10} ϑ_{2} - 2 b^{2} α_{1} α_{4} α_{8} α_{10} ϑ_{2} + b^{2} α_{1} α_{5} α_{8} α_{10} ϑ_{2} + b^{2} α_{3} α_{5} α_{8} α_{10} ϑ_{2} + α_{1} α_{3} α_{5} α_{8} α_{10} ϑ_{2}

+ b^{2} α_{4} α_{6} α_{8} α_{10} ϑ_{2} + α_{1} α_{4} α_{6} α_{8} α_{10} ϑ_{2} - b^{6} α_{1} α_{4} δ_{1} ϑ_{2} - b^{4} α_{1} α_{5} α_{7} δ_{1} ϑ_{2} - b^{6} α_{4} α_{10} δ_{1} ϑ_{2} - 3 b^{4} α_{1} α_{4} α_{10} δ_{1} ϑ_{2}

- b^{4} α_{5} α_{7} α_{10} δ_{1} ϑ_{2} - 2 b^{2} α_{1} α_{5} α_{7} α_{10} δ_{1} ϑ_{2} - b^{4} α_{1} α_{4} α_{10} δ_{3} ϑ_{2} + b^{2} α_{1} α_{3} α_{5} α_{10} δ_{3} ϑ_{2} + b^{2} α_{1} α_{4} α_{6} α_{10} δ_{3} ϑ_{2}

(58)

+ b^{4} α_{5} α_{7} ϵ_{3} ϑ_{2} - b^{4} α_{4} α_{8} ϵ_{3} ϑ_{2} + 2 b^{2} α_{5} α_{7} α_{10} ϵ_{3} ϑ_{2} - 2 b^{2} α_{4} α_{8} α_{10} ϵ_{3} ϑ_{2} - b^{4} α_{4} α_{10} δ_{3} ϵ_{3} ϑ_{2} - b^{4} α_{5} ϵ_{3} ϵ_{7} ϑ_{2} - b^{2} α_{5} α_{8} ϵ_{3} ϵ_{7} ϑ_{2} - 2 b^{2} α_{5} α_{10} ϵ_{3} ϵ_{7} ϑ_{2} - α_{5} α_{8} α_{10} ϵ_{3} ϵ_{7} ϑ_{2} - b^{2} α_{5} α_{10} δ_{3} ϵ_{3} ϵ_{7} ϑ_{2}) .

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